Analysis and Geometry in Metric Spaces: Sobolev Mappings, the Heisenberg Group, and the Whitney Extension TheoremZimmerman, Scott (2017) Analysis and Geometry in Metric Spaces: Sobolev Mappings, the Heisenberg Group, and the Whitney Extension Theorem. Doctoral Dissertation, University of Pittsburgh. (Unpublished) This is the latest version of this item.
AbstractThis thesis focuses on analysis in and the geometry of the Heisenberg group as well as geometric properties of Sobolev mappings. It begins with a detailed introduction to the Heisenberg group. After, we see a new and elementary proof for the structure of geodesics in the subRiemannian Heisenberg group. We also prove that the CarnotCarath\'{e}odory metric is real analytic away from the center of the group. Next, we prove a version of the classical Whitney Extension Theorem for curves in the Heisenberg group. Given a real valued function defined on a compact set in Euclidean space, the classical Whitney Extension Theorem from 1934 gives necessary and sufficient conditions for the existence of a $C^k$ extension defined on the entire space. We prove a version of the Whitney Extension Theorem for $C^1$, horizontal curves in the Heisenberg group. We then turn our attention to Sobolev mappings. Finally, we generalize a classical result of Dubovitski\u{\i} for smooth maps to the case of Sobolev mappings. In 1957, Duvovitski\u{\i} generalized Sard's classical theorem by establishing a bound on the Hausdorff dimension of the intersection of the critical set of a smooth map and almost every one of its level sets. We show that Dubovitski\u{\i}'s theorem Share
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